Lesson%2011.pdf

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Module

5Knowledge

Representation and

Logic (Propositional Logic)

Version 2 CSE IIT, Kharagpur

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5.1 Instructional Objective

Students should understand the importance of knowledge representation in intelligentagents

Students should understand the use of formal logic as a knowledge representationlanguage

The student should be familiar with the following concepts of logico syntaxo semanticso validityo satisfiabilityo interpretation and modelso entailment

Students should understand each of the above concepts in propositional logic Students should learn different inference mechanisms in propositional logicAt the end of this lesson the student should be able to do the following:

Represent a natural language description as statements in logic Deduct new sentences by applying inference rules.


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Lesson11

Propositional Logic


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5.2 Knowledge Representation and Reasoning

Intelligent agents should have capacity for:

Perceiving, that is, acquiring information from environment,

Knowledge Representation, that is, representing its understanding of the world, Reasoning, that is, inferring the implications of what it knows and of the choices

it has, and Acting, that is, choosing what it want to do and carry it out.

Representation of knowledge and the reasoning process are central to the entire field ofartificial intelligence. The primary component of a knowledge-based agent is itsknowledge-base. A knowledge-base is a set of sentences. Each sentence is expressed in alanguage called the knowledge representation language. Sentences represent someassertions about the world. There must mechanisms to derive new sentences from oldones. This process is known as inferencing or reasoning. Inference must obey the primary

requirement that the new sentences should follow logically from the previous ones.

Logic is the primary vehicle for representing and reasoning about knowledge.Specifically, we will be dealing with formal logic. The advantage of using formal logic asa language of AI is that it is precise and definite. This allows programs to be writtenwhich are declarative - they describe what is true and not how to solve problems. Thisalso allows for automated reasoning techniques for general purpose inferencing.

This, however, leads to some severe limitations. Clearly, a large portion of the reasoningcarried out by humans depends on handling knowledge that is uncertain. Logic cannotrepresent this uncertainty well. Similarly, natural language reasoning requires inferring

hidden state, namely, the intention of the speaker. When we say, "One of the wheel of thecar is flat.", we know that it has three wheels left. Humans can cope with virtually infinitevariety of utterances using a finite store of commonsense knowledge. Formal logic hasdifficulty with this kind of ambiguity.

A logic consists of two parts, a language and a method of reasoning. The logicallanguage, in turn, has two aspects, syntax and semantics. Thus, to specify or define aparticular logic, one needs to specify three things:

Syntax:The atomic symbols of the logical language, and the rules for constructing well-formed, non-atomic expressions (symbol structures) of the logic. Syntax specifies thesymbols in the language and how they can be combined to form sentences. Hence factsabout the world are represented as sentences in logic.

Semantics: The meanings of the atomic symbols of the logic, and the rules fordetermining the meanings of non-atomic expressions of the logic. It specifies what factsin the world a sentence refers to. Hence, also specifies how you assign a truth value to asentence based on its meaning in the world. A factis a claim about the world, and may betrue or false.


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Syntactic Inference Method: The rules for determining a subset of logical expressions,called theorems of the logic. It refers to mechanical method for computing (deriving) new(true) sentences from existing sentences.

Factsare claims about the world that are True or False, whereas a representationis an

expression (sentence) in some language that can be encoded in a computer program andstands for the objects and relations in the world. We need to ensure that the representationis consistent with reality, so that the following figure holds:

ent ai l sRepr esent at i on: Sent ences - - - - - - - - - - - - - - > Sent ences

| || || Semant i cs | Semant i cs| ref er to | ref er t o| |\ / f ol l ows \ /

Wor l d: Facts - - - - - - - - - - - - - - - - - - > Facts

There are a number of logical systems with different syntax and semantics. We list belowa few of them.

Propositional logicAll objects described are fixed or unique

"John is a student" student(john)

Here John refers to one unique person.

First order predicate logicObjects described can be unique or variables to stand for a unique object

"All students are poor"

ForAll(S) [student(S) -> poor(S)]

Here S can be replaced by many different unique students.

This makes programs much more compact:

eg. ForAll(A,B)[brother(A,B) -> brother (B,A)]

replaces half the possible statements about brothers


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TemporalRepresents truth over time.

ModalRepresents doubt

Higher order logicsAllows variable to represent many relations between objects

Non-monotonicRepresents defaults

Propositional is one of the simplest systems of logic.

5.3 Propositional Logic

In propositional logic (PL) an user defines a set of propositional symbols, like Pand Q.User defines the semantics of each of these symbols. For example,

o P means "It is hot"o Q means "It is humid"o R means "It is raining"o

A

sentence(also called a formula or well-formed formula or wff) is defined as:1. A symbol

2. If S is a sentence, then ~S is a sentence, where "~" is the "not" logicaloperator

3. If S and T are sentences, then (S v T), (S ^ T), (S => T), and (S T) aresentences, where the four logical connectives correspond to "or," "and,""implies," and "if and only if," respectively

4. A finite number of applications of (1)-(3)

Examples of PL sentences:o (P ^ Q) => R (here meaning "If it is hot and humid, then it is raining")o Q => P (here meaning "If it is humid, then it is hot")o Q (here meaning "It is humid.")


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Given the truth values of all of the constituent symbols in a sentence, thatsentence can be "evaluated" to determine its truth value (True or False). This iscalled an interpretationof the sentence.

A modelis an interpretation (i.e., an assignment of truth values to symbols) of aset of sentences such that each sentence is True. A model is just a formalmathematical structure that "stands in" for the world.

A valid sentence (also called a tautology) is a sentence that is True under allinterpretations. Hence, no matter what the world is actually like or what thesemantics is, the sentence is True. For example "It's raining or it's not raining."

An inconsistent sentence (also called unsatisfiable or a contradiction) is asentence that is False under allinterpretations. Hence the world is never like whatit describes. For example, "It's raining and it's not raining."

Sentence P entailssentence Q, written P |= Q, means that whenever P is True, sois Q. In other words, all models of P are also models of Q

Example: Entailment

Show that:

Proof: For any model M in which holds then we know that pholds in M

and holds in M. Sincepholds in M then since holds in M, qmust hold in

M. Therefore qholds in every model that holds and so .

As we have noted models affect equivalence and so we repeat the definition again and

give an example of a proof of equivalence.


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Example: Equivalence

Show that:

Proof: We need to provide two proofs as above for

For any model M in which holds then we know that either holds in Mand so holds in M, or does not hold in M and so holds in M. Since either

holds in M or holds in M, then holds in M.

and

For any model M in which holds then we know that either holds in Mor holds in M. If holds in M then holds in M. Otherwise, if holds

in M then holds in M. Therefore holds in M.

Knowledge based programming relies on concluding new knowledge from existing

knowledge. Entailment is a required justification; i.e. if is known then there is

justification to conclude if

In some circumstances we insist on this strong form of justification; i.e. we cannot

conclude unless the entailment holds. Reasoning like this is the equivalent forknowledge based programs of running a piece of conventional software.

Note:Entailment ( ) is concerned with truthand is determined by considering the truthof the sentences in all models.


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5.4 Propositional Logic Inference

Let KB = { S1, S2,..., SM } be the set of all sentences in our Knowledge Base, whereeach Si is a sentence in Propositional Logic. Let { X1, X2, ..., XN } be the set of all thesymbols (i.e., variables) that are contained in all of the M sentences in KB. Say we want

to know if a goal (aka query, conclusion, or theorem) sentence G follows from KB.

5.4.1 Model Checking

Since the computer doesn't know the interpretation of these sentences in the world, wedon't know whether the constituent symbols represent facts in the world that are True orFalse. So, instead, consider allpossible combinations of truth values for all the symbols,hence enumerating all logically distinct cases:

X1 X2 . . . XN | S1 S2 . . . SM | S1 ^ S2 . . . ^ SM | G | ( S1 . . . ^ SM) => G- - - - - - - - - - - - - | - - - - - - - - - - - - - - | - - - - - - - - - - - - - - - - - - | - - - | - - - - - - - - - - - - - - - - - - -F F . . . F | | | |F F . . . T | | | |. . . | | | |T T . . . T | | | |

There are 2^N rows in the table. Each row corresponds to an equivalence class of worlds that, under a given

interpretation, have the truth values for the N symbols assigned in that row. The modelsof KB are the rows where the third-to-last column is true, i.e., where

all of the sentences in KB are true. A sentence R is validif and only if it is true under all possible interpretations, i.e.,

if the entire column associated with R contains all truevalues.

Since we don't know the semantics and therefore whether each symbol is True orFalse, to determine if a sentence G is entailedby KB, we must determine if allmodels of KB are also models of G. That is, whenever KB is true, G is true too. Inother words, whenever the third-to-last column has a T, the same row in thesecond-to-last column also has a T. But this is logically equivalent to saying thatthe sentence (KB => G) is valid (by definition of the "implies" connective). Inother words, if the last column of the table above contains only True, then KBentails G; or conclusion G logically follows from the premises in KB, no matterwhat the interpretations (i.e., semantics) associated with all of the sentences!

The truth table method of inference is complete for PL (Propositional Logic)because we can always enumerate all 2^n rows for the npropositional symbols

that occur. But this is exponential in n. In general, it has been shown that theproblem of checking if a set of sentences in PL is satisfiable is NP-complete. (Thetruth table method of inference is notcomplete for FOL (First-Order Logic).)

Example

Using the "weather" sentences from above, let KB = (((P ^ Q) => R) ^ (Q => P) ^ Q)


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corresponding to the three facts we know about the weather: (1) "If it is hot and humid,then it is raining," (2) "If it is humid, then it is hot," and (3) "It is humid." Now let's askthe query "Is it raining?" That is, is the query sentence R entailed by KB? Using the truth-table approach to answering this query we have:

P Q R | ( P ^ Q) => R | Q => P | Q | KB | R | KB => R- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -T T T T T T T T TT T F F T T F F TT F T T T F F T TT F F T T F F F TF T T T F T F T TF T F T F T F F TF F T T T F F T TF F F T T F F F T

Hence, in this problem there is only one model of KB, when P, Q, and R are all True.And in this case R is also True, so R is entailed by KB. Also, you can see that the lastcolumn is all True values, so the sentence KB => R is valid.

Instead of an exponential length proof by truth table construction, is there a faster way toimplement the inference process? Yes, using a proof procedureor inference procedurethat uses sound rules of inferenceto deduce (i.e., derive) new sentences that are true inall cases where the premises are true. For example, consider the following:

P Q | P P => Q | P ^ ( P => Q) | Q | ( P ^ ( P => Q) ) => Q- - - - - - | - - - - - - - - - - - - | - - - - - - - - - - - - - - | - - - - - - - - - - - - - - - - - - - - - - - - -F F | F T | F | F | TF T | F T | F | T | TT F | T F | F | F | TT T | T T | T | T | T

Since whenever P and P => Q are both true (last row only), Q is true too, Q is said to bederivedfrom these two premise sentences. We write this as KB |- Q. This local patternreferencing only two of the M sentences in KB is called the Modus Ponens inferencerule. The truth table shows that this inference rule is sound. It specifies how to make onekind of step in deriving a conclusion sentence from a KB.

Therefore, given the sentences in KB, construct a proofthat a given conclusion sentencecan be derived from KB by applying a sequence of sound inferences using either

sentences in KB or sentences derived earlier in the proof, until the conclusion sentence isderived. This method is called the Natural Deductionprocedure. (Note: This step-by-step, local proof process also relies on the monotonicityproperty of PL and FOL. Thatis, adding a new sentence to KB does not affect what can be entailed from the originalKB and does not invalidate old sentences.)


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